Author: Jameson O'Reilly

# QCJC: Cirac & Zoller 1994

Before anyone had even dreamed of quantum computing, trapped ions were used for atomic clocks and setting frequency standards. Their superb isolation from the environment let physicists make the world’s most precise and accurate measurements of energy level structures and frequencies. It also led to interest in using trapped ions as a quantum information platform, especially after the publication of Shor’s factoring algorithm sparked mainstream interest in the field in 1994.

Barely over a week later, PRL received a transcript from Cirac and Zoller detailing how to implement controlled-NOT gates on arbitrary numbers of ions in the same trap. These gates, along with single-qubit rotations around any axis, form a universal set of gates for quantum computation. This means that a sequence of them can approximate any unitary transformation to arbitrary accuracy. Since it was already known how to implement the single-qubit rotation gates, this paper established trapped ions as a candidate platform for universal QC.

In a typical ion trap, the ions are very tightly confined in the Y and Z directions and more loosely trapped in the X direction. This allows them to be trapped in a straight line, where repulsive Coulomb interactions keep them a few microns apart. Just like the internal levels of each ion, the collective movement of the ions in the trap is quantized. There are specific modes of motion that have discrete energies. The lowest-energy motion is the center-of-mass mode, which is when all the ions move together as one, back and forth, in the X direction.

The transition between internal energy states of an ion is driven by applying a laser with the correct frequency, and hence photon energy. Lowering the frequency by the energy gap between the no-phonon (not moving) $|0\rangle$ and one-phonon (center-of-mass mode) $|1\rangle$ states* gives the Hamiltonian

$\hat{H}_{n,q} = \frac{\eta}{\sqrt{N}} \frac{\Omega}{2} [|e_q \rangle_n \langle g| a e^{-i\phi} + |g \rangle_n \langle e_q| a^\dagger e^{i\phi} ]$

where $a$ and $a^\dagger$ are the phonon creation and annihilation operators, the $n$ subscript refers to the nth ion in the chain, and the $q$ subscript denotes the laser polarization and consequently which excited state the ion is pumped into.

Applying this Hamiltonian (by shining laser light on the relevant ion) for a time $t = \frac{k \pi \sqrt{N}}{\Omega \eta}$, also known as the $k\pi$ time, evolves the state according to the unitary operator

$\hat{U}^{k,q}_{n} (\phi) = \exp(\frac{-i k \pi}{2} [|e_q \rangle_n \langle g| a e^{-i\phi} + |g \rangle_n \langle e_q| a^\dagger e^{i\phi} ])$

*Thanks to the detuning, this will leave the states $|g\rangle |0\rangle$ and $|e_q\rangle |1\rangle$ alone while swapping the states $|e_q\rangle |0\rangle$ and $|g\rangle |1\rangle$ for every $\pi$ rotation. When the states swap, they also pick up a phase of $-i$.

If we now consider any two ions that we want to entangle and cool the ion chain to its motional ground state, the four relevant states are $|g \rangle_m |g \rangle_n |0 \rangle$$|g \rangle_m |e_0 \rangle_n |0 \rangle$$|e_0 \rangle_m |g \rangle_n |0 \rangle$, and $|e_0 \rangle_m |e_0 \rangle_n |0 \rangle$. Applying a $\pi$ rotation to the mth ion leads to the changes

$|g\rangle_m |g\rangle_n |0\rangle \rightarrow |g\rangle_m |g\rangle_n |0\rangle$

$|g\rangle_m |e_0\rangle_n |0\rangle \rightarrow |g\rangle_m |e_0\rangle_n |0\rangle$

$|e_0\rangle_m |g\rangle_n |0\rangle \rightarrow -i|g\rangle_m |g\rangle_n |1\rangle$

$|e_0\rangle_m |e_0\rangle_n |0\rangle \rightarrow -i|g\rangle_m |e_0\rangle_n |1\rangle$

The two states with $|e_0\rangle$ changed and picked up phases. The next step in the protocol involves a $2\pi$ rotation applied to the nth ion. This will leave the fourth state alone but apply two swaps and two phases to the third state, ultimately just applying a phase of $-i * -i = -1$. This would leave only one state (the fourth) with a minus sign if not for the fact that a $\pi$ rotation will also affect the second state. To avoid this, we can use polarization that couples the $|g\rangle$ state to $|e_1\rangle$ instead of $|e_0\rangle$. That way, the third state can pick up its phase while the second state remains unaffected due to the fact that $|e_0\rangle$ and $|e_1\rangle$ do not couple through this interaction.

Finally, reapplying a $\pi$ rotation on the mth ion yields the transformations

$|g\rangle_m |g\rangle_n |0\rangle \rightarrow |g\rangle_m |g\rangle_n |0\rangle$

$|g\rangle_m |e_0\rangle_n |0\rangle \rightarrow |g\rangle_m |e_0\rangle_n |0\rangle$

$i|g\rangle_m |g\rangle_n |0\rangle \rightarrow |e_0\rangle_m |g\rangle_n |0\rangle$

$-i|g\rangle_m |e_0\rangle_n |0\rangle \rightarrow -|e_0\rangle_m |e_0\rangle_n |0\rangle$

Thus, there is a phase change only if both ions were initially excited. That this represents a CNOT gate may not be obvious, but if we take the usual definition $|\pm\rangle = \frac{|g\rangle + |e_0\rangle}{\sqrt{2}}$, then the above transformations can be summarized as

$|g\rangle_m |\pm\rangle_n \rightarrow |g\rangle_m |\pm\rangle_n$

$|e_0\rangle_m |\pm\rangle_n \rightarrow |e_0\rangle_m |\mp\rangle_n$

Applying the right X rotations on the nth ion make this transformation what we typically think of when we say CNOT, although it may be more efficient to leave it be.

After going through the implementation of this gate, Cirac and Zoller spend a few paragraphs on a numerical simulation of a trapped ion quantum computer carrying out a Quantum Fourier Transform, the main ingredient in Shor’s algorithm. Only a few months later, the first CNOT gate was implemented experimentally and many of the existing trapped ion research groups turned towards quantum computing, at least in some capacity. Today, we are inching closer to the dream this paper sparked in 1995, that of a universal, trapped-ion quantum computer.

Among potential experimental realizations of scalable quantum computing (QC), superconducting qubits of various kinds have received the bulk of the public’s attention. In addition to their promise, this is largely because IBM and Google have both chosen to pour money and resources into developing superconducting devices for QC. As massive tech companies, they are both well-known to the public and have some experience in developing and scaling the fabrication methods used to make superconducting circuits. This expertise made superconducting their natural technology of choice and since then has stimulated rapid progress, another reason for so much press coverage.

Meanwhile, trapped ions, another candidate for scalable QC, has reached a similar level of maturity while receiving much less attention. Instead of the “artificial atoms” engineered in superconducting circuits, trapped ions do computation using the energy levels of an actual atom that has been ionized so that it can be trapped by electric (and/or magnetic, depending on your trap architecture) fields.

One example of such a qubit is the two hyperfine-split ground state levels of an Ytterbium-171 ion. Hyperfine splitting of an energy level is due to the interaction between the total electron angular momentum and the nuclear spin. Yb-171 has a nuclear spin of 1/2 so each of its energy levels, including the ground state, is split into two different levels, one for spin up and one for spin down. By labeling one of these states |0> and the other |1>, we have ourselves a qubit. I won’t talk much more about how these qubits are implemented and controlled here because hopefully I’ll cover that in future posts.

Instead, I figured there would be no better way to start off my QCJC contributions than by comparing trapped ions with superconducting qubits. Thus far, QCJC (at least the experimental posts) has predominantly been about superconducting qubits because, duh, Chunny works in a superconducting lab. Norbert Linke and Chris Monroe, on the other hand, do not. Instead, Chris Monroe is the PI of a trapped ion group at UMD and Norbert Linke is a research scientist in that group.

Last year, along with collaborators from inside and outside their group, they published a comparison of a five-qubit trapped-ion quantum computer that they built with the five-qubit superconducting quantum computer made publicly available online by IBM. Both systems are full-stack, fully-programmable universal quantum computers, but they are also both noisy. This means that their qubits decohere over time and their gates are not perfectly implemented.

To avoid decoherence, it is important that gate times are much shorter than the decoherence time so that all of the gates required for an algorithm can be performed without having to worry about the quality of the qubit degrading. Superconducting devices decohere through state relaxation (T1 = 60 us) and dephasing (T2* = 60 us) while trapped ions only decohere via dephasing (T2* = 0.5 s). Although superconducting qubits decohere much more quickly than hyperfine qubits do, they also have much faster gate times at 130 ns for one-qubit gates and between 250 and 450 ns for two-qubit gates compared to 20 us and 250 us for one- and two-qubit gates, respectively.

Gate time is not the only consideration for avoiding decoherence since different numbers of gates may be needed in each architecture due to different sets being available. Trapped ions can implement arbitrary rotations on single qubits and XX entangling gates on two qubits. IBM makes the Clifford+T gate set available in their interface and have a compiler that optimizes the actual implementation of whatever algorithm one gives. Using fewer gates decreases the total time needed for the calculation, thus preventing more decoherence, but it also decreases the impact of gate errors because there are less chances for them to occur.

The final important difference between the two architectures is their level of connectivity. Currently, not all superconducting qubits on the IBM chip are able to interact directly because they are confined to a planar geometry and a direct connection is need to apply a two-qubit gate. two non-connected qubits can still become entangled by using an intermediary qubit, but this requires the use of extra gates, which takes time and gives more opportunities for errors to creep in. In contrast, any two ions in the same trap can interact by using the collective motion of all the ions as an information bus (I need to read the paper that proposes this and write a QCJC submission on it in the future because I don’t really understand it).

With all of this in mind, Linke and his team first implemented a Margolus gate on both systems. This gate is the same as a Toffoli gate except that it introduces a phase on the |001> state and can be implemented with fewer elementary gates. In particular, the star-shaped connectivity graph of the superconducting system does not require any extra gates to implement it. The IBM system completed this circuit with a success probability of 74.1(7)% and the trapped ion system had a success probability of 90.1(2)%. For full Toffoli gates, the success probabilities were 52.6(8)% and 85.0(2)%. All of these figures are based on state tomography done after the gates were implemented.

Next, they ran an algorithm to find the c of a black box that implements a function f(x) = x • c. Known as the Bernstein-Vazirani problem, this requires multiple queries to the function oracle for classical systems but can be completed in one query by a quantum computer. This particular algorithm maps well onto the star-shaped connectivity architecture of the IBM chip. The success probabilities for completing this algorithm were 72.8(5)% for superconducting and 85.1(1)% for trapped ions.

Finally, the two systems were made to solve the hidden shift problem, in which an oracle calculates a known function f(x) but with a “hidden shift”: f(x + s). The idea of the algorithm is to find this hidden shift s. This algorithm is not nearly so kind to IBM’s connectivity graph, resulting in a success probability of only 35.1(6)% compared to 77.1(2) for the trapped ion quantum computer.

Overall, this may seem like a victory for trapped ions, but we have no way of knowing how these comparisons will change as both systems are further developed and scaled up. Neither has hit any fundamental obstacles to scaling yet. Both are still promising technologies with their own strengths and weaknesses and there is a reason that Google and IBM have chosen to focus on superconducting circuits. With their expertise in fabrication and scaling technologies, rapid progress that outpaces trapped ions could be on the horizon. Or, it might be the case that using the two together, or one with some other technology, will prove to be the architecture of the future.

Source: Linke, N. M., et al. Experimental Comparison of Two Quantum Computing Architectures PNAS 114 (13) 3305-3310 (2017)